Optimal. Leaf size=38 \[ -\frac {2 a c^4 (a-b x)^5}{5 b}+\frac {c^4 (a-b x)^6}{6 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45}
\begin {gather*} \frac {c^4 (a-b x)^6}{6 b}-\frac {2 a c^4 (a-b x)^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x) (a c-b c x)^4 \, dx &=\int \left (2 a (a c-b c x)^4-\frac {(a c-b c x)^5}{c}\right ) \, dx\\ &=-\frac {2 a c^4 (a-b x)^5}{5 b}+\frac {c^4 (a-b x)^6}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 68, normalized size = 1.79 \begin {gather*} c^4 \left (a^5 x-\frac {3}{2} a^4 b x^2+\frac {2}{3} a^3 b^2 x^3+\frac {1}{2} a^2 b^3 x^4-\frac {3}{5} a b^4 x^5+\frac {b^5 x^6}{6}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs.
\(2(34)=68\).
time = 0.06, size = 73, normalized size = 1.92
method | result | size |
gosper | \(\frac {x \left (5 b^{5} x^{5}-18 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} b^{2} x^{2}-45 a^{4} b x +30 a^{5}\right ) c^{4}}{30}\) | \(59\) |
default | \(\frac {1}{6} b^{5} c^{4} x^{6}-\frac {3}{5} a \,b^{4} c^{4} x^{5}+\frac {1}{2} a^{2} b^{3} c^{4} x^{4}+\frac {2}{3} a^{3} b^{2} c^{4} x^{3}-\frac {3}{2} a^{4} b \,c^{4} x^{2}+a^{5} c^{4} x\) | \(73\) |
norman | \(\frac {1}{6} b^{5} c^{4} x^{6}-\frac {3}{5} a \,b^{4} c^{4} x^{5}+\frac {1}{2} a^{2} b^{3} c^{4} x^{4}+\frac {2}{3} a^{3} b^{2} c^{4} x^{3}-\frac {3}{2} a^{4} b \,c^{4} x^{2}+a^{5} c^{4} x\) | \(73\) |
risch | \(\frac {1}{6} b^{5} c^{4} x^{6}-\frac {3}{5} a \,b^{4} c^{4} x^{5}+\frac {1}{2} a^{2} b^{3} c^{4} x^{4}+\frac {2}{3} a^{3} b^{2} c^{4} x^{3}-\frac {3}{2} a^{4} b \,c^{4} x^{2}+a^{5} c^{4} x\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 72, normalized size = 1.89 \begin {gather*} \frac {1}{6} \, b^{5} c^{4} x^{6} - \frac {3}{5} \, a b^{4} c^{4} x^{5} + \frac {1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac {2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac {3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 72, normalized size = 1.89 \begin {gather*} \frac {1}{6} \, b^{5} c^{4} x^{6} - \frac {3}{5} \, a b^{4} c^{4} x^{5} + \frac {1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac {2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac {3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (29) = 58\).
time = 0.02, size = 82, normalized size = 2.16 \begin {gather*} a^{5} c^{4} x - \frac {3 a^{4} b c^{4} x^{2}}{2} + \frac {2 a^{3} b^{2} c^{4} x^{3}}{3} + \frac {a^{2} b^{3} c^{4} x^{4}}{2} - \frac {3 a b^{4} c^{4} x^{5}}{5} + \frac {b^{5} c^{4} x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 72, normalized size = 1.89 \begin {gather*} \frac {1}{6} \, b^{5} c^{4} x^{6} - \frac {3}{5} \, a b^{4} c^{4} x^{5} + \frac {1}{2} \, a^{2} b^{3} c^{4} x^{4} + \frac {2}{3} \, a^{3} b^{2} c^{4} x^{3} - \frac {3}{2} \, a^{4} b c^{4} x^{2} + a^{5} c^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 72, normalized size = 1.89 \begin {gather*} a^5\,c^4\,x-\frac {3\,a^4\,b\,c^4\,x^2}{2}+\frac {2\,a^3\,b^2\,c^4\,x^3}{3}+\frac {a^2\,b^3\,c^4\,x^4}{2}-\frac {3\,a\,b^4\,c^4\,x^5}{5}+\frac {b^5\,c^4\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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